Cone Surface Area Calculator
Cone total surface, volume, and slant height.
What it does
Cones wrap a circle into a point. Total surface adds the base disk to the side that unrolls into a sector. Volume is one third of the matching cylinder, because three cones fill that cylinder. One radius and one height fix slant, surface, and volume together.
The tool above runs the trio. With radius 5 cm and height 12 cm, slant is 13 cm, surface is 282.74 cm2, and volume is 314.2 cm3. The imperial mirror at 2 in by 4.7 in gives 44.66 in2, 19.7 in3, and a 5.11 in slant. Funnels read surface while hoppers read volume.
The 5-12-13 triangle is the classic check. Slant follows Pythagoras on radius and height, then surface multiplies pi by radius times radius plus slant. That is why small radius errors matter twice: once in the base circle and once along the side.
This page assumes a right circular cone with a flat base. It does not cover oblique tips, elliptical bases, or wall thickness. For an open funnel, subtract the base circle from the total shown.
How to use
- Pick metric (cm) or imperial (in).
- Enter the base radius, not the diameter.
- Enter the vertical height.
- Read slant height to check the triangle.
- Read total surface, then volume.
- Halve the diameter entry if measured across.
Formula
- r
- Radius of the circular base, in cm or in.
- h
- Vertical height from base to tip, in the same unit.
- s
- Slant edge from rim to tip.
Right circular cone with a flat circular base. At height zero the solid flattens to a disk; at radius zero there is no solid. Open cones drop the base circle.
Worked example
Example: radius 5 cm, height 12 cm
Defaults: metric radius 5 cm with height 12 cm. Imperial mirror is 2 in by 4.7 in.
- Inputs (metric default): radius 5 cm, height 12 cm.
- Slant: root of 25 plus 144 is 13 cm.
- Surface: pi times 5 times 18 is 282.74 cm2.
- Volume: pi times 25 times 12 / 3 is 314.2 cm3.
- Imperial mirror (2 in, 4.7 in) gives 44.66 in2, 19.7 in3, 5.11 in. Tool shows the same.
Reference table
| Size | Surface | Volume |
|---|---|---|
| 5 cm by 12 cm | 282.74 cm2 | 314.2 cm3 |
| 3 cm by 4 cm | 75.4 cm2 | 37.7 cm3 |
| 4 cm by 3 cm | 113.1 cm2 | 50.3 cm3 |
| 2 in by 4.7 in | 44.66 in2 | 19.7 in3 |
| 6 cm by 8 cm | 301.59 cm2 | 301.6 cm3 |
Limits
- Right circular cones only. Oblique needs other math.
- Thin wall assumed. Thick walls need inner and outer passes.
- Flat base. Domed bases add area.
- No waste or seams. Sheet work needs extra.
Common mistakes
- Entering diameter as radius, which quadruples the base.
- Mixing radius units with height units.
- Forgetting the base on a closed cone.
- Using slant as vertical height.
- Rounding pi early on large cones.
Try it yourself
Quick checks. Answers sit tucked under each question.
Radius 3 cm, height 4 cm?
Slant is root of 9 plus 16 is 5 cm. Surface is pi times 3 times 8 is 75.4 cm2. Volume is pi times 9 times 4 / 3 is 37.7 cm3.
Radius 4 cm, height 3 cm?
Slant is root of 16 plus 9 is 5 cm. Surface is pi times 4 times 9 is 113.1 cm2. Volume is pi times 16 times 3 / 3 is 50.3 cm3.
Questions
What is the surface formula?
SA = pi r squared plus pi r s, or pi r (r plus s). Here pi times 5 times 18 is 282.74 cm2.
How is volume figured?
V = pi r squared h / 3. Here pi times 25 times 12 / 3 is 314.2 cm3.
How is slant height figured?
Slant s = root of r squared plus h squared. Here root 169 is 13 cm.
Does this work for tilted cones?
No. This is a right circular cone. Oblique or elliptical cones need other methods.
Can I mix cm and inches?
Use one unit per calculation. Convert first if mixed.
What happens if I double the size?
Surface squares and volume cubes. Double both and surface quadruples while volume grows eightfold.